Asymptotic behavior of functions. comparison of infinitesimal functions

As noted in previous section, the study of classical algorithms in many cases can be carried out using asymptotic methods of mathematical statistics, in particular, using CLT and convergence inheritance methods. The separation of classical mathematical statistics from the needs of applied research manifested itself, in particular, in the fact that popular monographs lack the mathematical apparatus necessary, in particular, for the study of two-sample statistics. The bottom line is that you have to go to the limit not by one parameter, but by two - the volumes of two samples. I had to develop an appropriate theory - the theory of inheritance of convergence, set out in our monograph.

However, the results of such a study will have to be applied with finite sample sizes. There is a whole bunch of problems associated with such a transition. Some of them were discussed in connection with the study of the properties of statistics constructed from samples from specific distributions.

However, when discussing the influence of deviations from initial assumptions on the properties of statistical procedures, additional problems arise. What deviations are considered typical? Should one focus on the most "harmful" deviations that distort the properties of algorithms to the greatest extent, or should one focus on "typical" deviations?

With the first approach, we get a guaranteed result, but the "price" of this result may be unnecessarily high. As an example, we point to the universal Berry-Esseen inequality for the error in the CLT. Quite rightly emphasizes A.A. Borovkov that "the rate of convergence in real problems, as a rule, turns out to be better."

In the second approach, the question arises which deviations are considered "typical". You can try to answer this question by analyzing large arrays of real data. It is quite natural that the answers of different research groups will differ, as can be seen, for example, from the results presented in the article.

One of the false ideas is the use in the analysis of possible deviations of only any specific parametric family - the Weibull-Gnedenko distributions, the three-parameter family of gamma distributions, etc. Back in 1927, acad. USSR Academy of Sciences S.N. Bernstein discussed the methodological error of reducing all empirical distributions to a four-parameter Pearson family. However, parametric methods of statistics are still very popular, especially among applied scientists, and the fault for this misconception lies primarily with teachers of statistical methods (see below, as well as the article).

15. Choosing one of many criteria to test a particular hypothesis

In many cases, many methods have been developed to solve a specific practical problem, and a specialist in mathematical research methods faces a problem: which one should be offered to an applied person for analyzing specific data?

As an example, consider the problem of checking the homogeneity of two independent samples. As you know, for its solution, you can offer a lot of criteria: Student, Cramer-Welch, Lord, chi-square, Wilcoxon (Mann-Whitney), Van - der - Waerden, Savage, N.V. Smirnov, such as omega-square (Lehmann -Rosenblatt), G.V. Martynova and others. Which one to choose?

The idea of ​​"voting" naturally comes to mind: to test by many criteria, and then decide "by a majority of votes". From the point of view of statistical theory, such a procedure simply leads to the construction of another criterion, which is a priori no better than the previous ones, but is more difficult to study. On the other hand, if the solutions are the same for all considered statistical criteria based on different principles, then, in accordance with the concept of stability, this increases the confidence in the overall solution obtained.

There is a widespread, especially among mathematicians, false and harmful opinion about the need to search for optimal methods, solutions, etc. The fact is that optimality usually disappears when there is a deviation from the initial assumptions. Thus, the arithmetic mean as an estimate of the mathematical expectation is optimal only when the original distribution is normal, while a consistent estimate is always, if only the mathematical expectation exists. On the other hand, for any arbitrary method of estimation or testing of hypotheses, one can usually formulate the concept of optimality in such a way that the method under consideration becomes optimal - from this specially chosen point of view. Take, for example, the sample median as an estimate of the mathematical expectation. It is, of course, optimal, although in a different sense than the arithmetic mean (optimal for a normal distribution). Namely, for the Laplace distribution, the sample median is the maximum likelihood estimate, and therefore optimal (in the sense specified in the monograph).

The homogeneity criteria have been analyzed in a monograph. There are several natural approaches to comparing criteria - based on the asymptotic relative efficiency according to Bahadur, Hodges-Lehman, Pitman. And it turned out that each criterion is optimal with the corresponding alternative or a suitable distribution on the set of alternatives. At the same time, mathematical calculations usually use the shift alternative, which is relatively rare in the practice of analyzing real statistical data (in connection with the Wilcoxon criterion, this alternative was discussed and criticized by us in ). The result is sad - the brilliant mathematical technique demonstrated in , does not allow us to give recommendations for choosing a test for homogeneity when analyzing real data. In other words, from the point of view of the application worker, i.e. analysis of specific data, the monograph is useless. Brilliant mastery of mathematics and great diligence demonstrated by the author of this monograph, alas, brought nothing to practice.

Of course, every practically working statistician in one way or another solves for himself the problem of choosing a statistical criterion. Based on a number of methodological considerations, we opted for the omega-square type criterion (Lehmann-Rosenblatt) that is consistent against any alternative. However, there is a feeling of dissatisfaction due to the insufficient validity of this choice.

Definition. The direction defined by a non-zero vector is called asymptotic direction relative to the second order line, if any the line of this direction (that is, parallel to the vector ) either has at most one common point with the line, or is contained in this line.

? How many common points can a line of the second order and a straight line of asymptotic direction relative to this line have?

In the general theory of second-order lines, it is proved that if

Then the non-zero vector ( defines the asymptotic direction with respect to the line

(general criterion for asymptotic direction).

For second order lines

if , then there are no asymptotic directions,

if then there are two asymptotic directions,

if then there is only one asymptotic direction.

The following lemma turns out to be useful ( criterion for the asymptotic direction of a line of parabolic type).

Lemma . Let be a line of parabolic type.

A non-zero vector has an asymptotic direction

relatively . (5)

(Problem. Prove the lemma.)

Definition. The straight line of asymptotic direction is called asymptote lines of the second order, if this line either does not intersect with or is contained in it.

Theorem . If has an asymptotic direction with respect to , then the asymptote parallel to the vector is determined by the equation

We fill in the table.

TASKS.

1. Find the asymptotic direction vectors for the following second order lines:

4 - hyperbolic type, two asymptotic directions.

Let us use the asymptotic direction criterion:

Has an asymptotic direction with respect to the given line 4 .

If =0, then =0, that is, zero. Then Divide by Get quadratic equation: , where t = . We solve this quadratic equation and find two solutions: t = 4 and t = 1. Then the asymptotic directions of the line .

(Two ways can be considered, since the line is of parabolic type.)

2. Find out if the coordinate axes have asymptotic directions relative to the lines of the second order:

3. Write the general equation of a second order line for which

a) the abscissa axis has an asymptotic direction;

b) Both coordinate axes have asymptotic directions;

c) the coordinate axes have asymptotic directions and O is the center of the line.

4. Write the asymptote equations for the lines:

a) ng w:val="EN-US"/>y=0"> ;

5. Prove that if a second-order line has two non-parallel asymptotes, then their intersection point is the center of this line.

Note: Since there are two non-parallel asymptotes, there are two asymptotic directions, then , and, therefore, the line is central.

Write the asymptote equations in general view and a system for finding the center. Everything is obvious.

6.(№920) Write the equation of a hyperbola passing through point A(0, -5) and having asymptotes x - 1 = 0 and 2x - y + 1 = 0.

indication. Use the statement of the previous problem.

Homework . , No. 915 (c, e, e), No. 916 (c, d, e), No. 920 (if you didn’t have time);

Cribs;

Silaev, Timoshenko. Practical tasks by geometry,

1 semester P.67, questions 1-8, p.70, questions 1-3 (oral).

SECOND-ORDER LINE DIAMETERS.

MATED DIAMETERS.

An affine coordinate system is given.

Definition. diameter line of the second order, conjugate to a vector of non-asymptotic direction with respect to , is the set of midpoints of all chords of the line parallel to the vector .

At the lecture, it was proved that the diameter is a straight line and its equation was obtained

Recommendations: Show (on an ellipse) how it is constructed (set a non-asymptotic direction; draw [two] straight lines of this direction intersecting the line; find the midpoints of the cut off chords; draw a straight line through the midpoints - this is the diameter).

Discuss:

1. Why is a vector of non-asymptotic direction taken in the definition of the diameter. If they cannot answer, then ask them to build a diameter, for example, for a parabola.

2. Does any line of the second order have at least one diameter? Why?

3. At the lecture it was proved that the diameter is a straight line. The middle of which chord is the point M in the figure?

4. Look at the brackets in equation (7). What do they remind?

Conclusion: 1) each center belongs to each diameter;

2) if there is a straight line of centers, then there is a single diameter.

5. What is the direction of the parabolic line diameters? (Asymptotic)

Proof (probably in a lecture).

Let the diameter d given by equation (7`) be conjugate to a vector of non-asymptotic direction. Then its direction vector

(-(), ). Let us show that this vector has an asymptotic direction. Let us use the criterion of the asymptotic direction vector for a parabolic line (see (5)). We substitute and make sure (do not forget that .

6. How many diameters does a parabola have? Their relative position? How many diameters do the rest of the parabolic lines have? Why?

7. How to construct the total diameter of some pairs of second-order lines (see questions 30, 31 below).

8. We fill in the table, be sure to make drawings.

1. . Write the equation for the set of midpoints of all chords parallel to the vector

2. Write an equation for the diameter d passing through the point K(1,-2) for the line.

Solution steps:

1st way.

1. Determine the type (to know how the diameters of this line behave).

In this case, the line is central, then all diameters pass through the center C.

2. We compose the equation of a straight line passing through two points K and C. This is the desired diameter.

2nd way.

1. We write the equation for the diameter d in the form (7`).

2. Substituting the coordinates of the point K into this equation, we find the relationship between the coordinates of the vector conjugate to the diameter d.

3. We set this vector, taking into account the found dependence, and compose the equation for the diameter d.

In this problem, it is easier to calculate in the second way.

3. . Write the equation for the diameter parallel to the x-axis.

4. Find the middle of the chord cut off by the line

on the line x + 3y – 12 =0.

Suggestion for a decision: Of course, you can find the points of intersection of the given line and line , and then - the middle of the resulting segment. The desire to do so disappears if we take, for example, a straight line with the equation x + 3y - 2009 = 0.

IN modern conditions interest in data analysis is constantly and intensively growing in completely different areas, such as biology, linguistics, economics, and, of course, IT. The basis of this analysis is statistical methods, and every self-respecting data mining specialist needs to understand them.

Unfortunately, really good literature, such that it would be able to provide both mathematically rigorous proofs and understandable intuitive explanations, is not very common. And these lectures, in my opinion, are unusually good for mathematicians who understand probability theory precisely for this reason. They are taught to masters at the German Christian-Albrecht University in the programs "Mathematics" and "Financial Mathematics". And for those who are interested in how this subject is taught abroad, I have translated these lectures. It took me several months to translate, I diluted the lectures with illustrations, exercises and footnotes to some theorems. I note that I am not a professional translator, but just an altruist and amateur in this field, so I will accept any criticism if it is constructive.

In short, the lectures are about:

Conditional expectation

This chapter does not deal directly with statistics, however, it is an ideal starting point for studying it. Conditional expectation is the best choice for predicting a random outcome based on the information you already have. And this is also random. Here, its various properties are considered, such as linearity, monotonicity, monotonic convergence, and others.

Point Estimation Basics

How to evaluate the distribution parameter? What is the criterion for this? What methods should be used for this? This chapter allows you to answer all these questions. Here the concepts of unbiased estimator and uniformly unbiased estimator with minimum variance are introduced. Explains where the chi-squared distribution and Student's distribution come from and why they are important in estimating the parameters of a normal distribution. It is told what Rao-Kramer's inequality and Fisher's information are. The concept of an exponential family is also introduced, which makes it many times easier to obtain a good estimate.

Bayesian and Minimax Parameter Estimation

A different philosophical approach to evaluation is described here. In this case, the parameter is considered unknown because it is a realization of some random variable with a known (a priori) distribution. Observing the result of the experiment, we calculate the so-called posterior distribution of the parameter. Based on this, we can get a Bayesian estimate, where the criterion is the minimum loss on average, or a minimax estimate, which minimizes the maximum possible loss.

Sufficiency and completeness

This chapter is of serious practical importance. A sufficient statistic is a function of the sample, such that it is sufficient to store only the result of this function in order to estimate the parameter. There are many such functions, and among them are the so-called minimal sufficient statistics. For example, to estimate the median of a normal distribution, it is enough to store only one number - the arithmetic mean over the entire sample. Does this also work for other distributions, like the Cauchy distribution? How do sufficient statistics help in choosing estimates? Here you can find answers to these questions.

Asymptotic properties of estimates

Perhaps the most important and necessary property of an estimate is its consistency, that is, the tendency to the true parameter with an increase in the sample size. This chapter describes the properties of the estimates known to us, obtained by the statistical methods described in the previous chapters. The concepts of asymptotic unbiasedness, asymptotic efficiency, and Kullback-Leibler distance are introduced.

Testing Basics

In addition to the question of how to evaluate a parameter unknown to us, we must somehow check whether it satisfies the required properties. For example, an experiment is being conducted in which a new drug is being tested. How do you know if you're more likely to get well with it than with older drugs? This chapter explains how such tests are built. You will learn what the uniformly most powerful test, the Neyman-Pearson test, significance level, confidence interval, and also where the notorious Gaussian test and t-test come from.

Asymptotic properties of criteria

Like estimates, criteria must satisfy certain asymptotic properties. Sometimes situations may arise when it is impossible to construct the required criterion, however, using the well-known central limit theorem, we construct a criterion that asymptotically tends to the necessary one. Here you will learn what the asymptotic significance level is, the likelihood ratio method, and how the Bartlett test and the chi-square independence test are built.

Linear model

This chapter can be considered as an addition, namely, the application of statistics in the case of linear regression. You will understand what grades are good and under what conditions. You will learn where the least squares method came from, how to build criteria and why you need an F-distribution.

The asymptotic behavior (or asymptotic behavior) of a function in the vicinity of a certain point a (finite or infinite) is understood as the nature of the change in the function as its argument x tends to this point. They usually try to represent this behavior with the help of another, simpler and more studied function, which, in the neighborhood of the point a, describes with sufficient accuracy the change in the function of interest to us or evaluates its behavior from one side or another. In connection with this, the problem arises of comparing the nature of the change of two functions in the neighborhood of the point a, which is connected with the consideration of their quotient. Of particular interest are the cases when both functions are either infinitely small (in.m.) or infinitely large (in.b.) for x and a. 10.1. Comparison of infinitesimal functions The main purpose of comparison of b.m. functions consists in comparing the nature of their approach to zero at x a, or the rate of their tending to zero. Let b.m. for x a the functions a(x) and P(x) are nonzero in some punctured neighborhood (a) of the point a, while at the point a they are equal to zero or undefined. Definition 10.1. The functions a(x) and 0(x) are called b.m. of the same order for a and write og (a:) = in O (/? («)) (the symbol O reads "Big O"), if for x a there is a non-zero finite limit of the ratio a (x) / /? ( i), i.e. It is obvious that then, according to (7.24), Zm € R \ (0), and the notation X ^ a0 [a (x)) is legal. The symbol O has the property of transitivity, i.e. if - indeed, taking into account Definition 10.1 and the property of the product of functions (see (7.23)), which have finite (in this case, not equal to zero) limits, we obtain the ASYMPTOTIC BEHAVIOR OF THE FUNCTIONS. Comparison of infinitesimal functions. Definition 10.2. The function a(x) they call b.m. of a higher order of smallness compared to (3 (x) (or relative to / 3 (x)) for x a and write) (the symbol o is read io small if the limit of the relation a exists and is equal to zero. In this case, also the function is said to be a bm of a lower order of smallness than a(x) for x a, and the word of smallness is usually omitted (as in the case of a higher order in Definition 10.2.) This means that if lim (then the function /)(x) is, according to definition 10.2, f.m. higher order compared to a(x) for x a and a(x) is b.m. lower order than /3(x) for x a, because in this case lijTi (fi(x)/ot(x)) . So we can write According to Theorem 7.3 on the connection between a function, its limit, and b.m. function from (10.3) it follows that ot) is a function, b.m. at. Hence a(x), i.e. values ​​|a(h)| for x close to a, many less values\0(x)\. In other words, the function a(x) tends to zero faster function/?(X). Theorem 10.1. The product of any b.m. for x a functions a(x) and P(x)) that are nonzero in some punctured neighborhood of the point a, there are b.m. a function of a higher order compared to each of the factors. Indeed, according to definition 10.2 b.m. of a higher order (taking into account Definition 7.10 of the b.m. of the function), the equalities mean the validity of the assertion of the theorem. Equalities containing the symbols O and o are sometimes called asymptotic estimates. Definition 10.3. The functions ot(x) and /3(x) are called incomparable b.m. for x -¥ a, if there is neither a finite nor an infinite limit of their ratio, i.e. if $ lim a(x)/0(x) (p £ is just like $ lim 0(x)/a(x)). Example 10.1. A. The functions a(x) = x and f(x) = sin2ar, by virtue of Definition 10.1, are b.m. of the same order at x 0, since taking into account (b. The function a (x) \u003d 1 - coss, by definition 10.2, - b. m. of a higher order compared to 0 (x) \u003d x at x 0, since with taking into account c. The function a(s) = \/x is a bm of a lower order compared to fl(x) = x for x 0, since r. The functions a(s) = x according to definition 10.3 are incomparable bm at x 0, since there is no limit (neither finite nor infinite - see example 7.5). x a b.m. of a higher order compared to xn~1) i.e. yapa \u003d ao (a: n "* 1), since lim (xL / xn" 1) \u003d If necessary, more accurate comparative characteristics behavior b.m. functions at x - and one of them is chosen as a kind of standard and called it the main one. Of course, the choice of the main b.m. to a certain extent arbitrary (they try to choose simpler: x for x - * 0; x-1 for x -41; 1 / x for x -\u003e oo, etc.). From the degrees 0k(x) the main b.m. functions f)(x) with different exponents k > 0 (when k ^ 0 0k(x) is not a f.m.) constitute a comparison sleeper for estimating a more complex f.m. functions a(z). Definition 10.4. The function a(z) is called b.m. k-th order of smallness with respect to (3(x) for x a, and the number k is the order of smallness if the functions a(z) and /3k(x) are b.m. of the same order for x a), i.e. if The word "small" is usually omitted in this case as well. Note: 1) the order k of one b.m. function relative to another can be any positive number; 2) if the order of the function a(x) relative to /3(x) is equal to k, then the order of the function P(x) with respect to a(x) is equal to 1/k, 3) it is not always possible to indicate a certain order of k for the bm function a(x), even if it is comparable with all powers of /?*(x). Example 10.2.a. The function cosx, according to definition 10.4, is b.m. of order k = 2 with respect to 0(x) = x at x 0, since taking into account b. Consider the functions. We will show that for any Indeed, according to ( 7.32).Thus, the function a1/1 is comparable with xk for any k > 0 for x -> 0, but it is not possible to indicate for this function the order of smallness with respect to x. with respect to the other is not always easy.We can recommend the following procedure: 1) write under the limit sign the ratio a(x) / 0k(x)\ 2) analyze the written ratio and try to simplify it; value of k) at which there will be a non-zero finite limit; 4) check the assumption by calculating the limit. Example 10.3. Let us determine the order of b.m. functions tgx - sin x with respect to x at x -» 0, i.e. let us find a number k > 0 such that we have an ASYMPTOTIC BEHAVIOR of the functions. Comparison of infinitesimal functions. At this stage, knowing that at x 0, according to (7.35) and (7.36), (sinx)/x 1 and cosx -> 1, and taking into account (7.23) and (7.33), we can determine that condition (10.7) will be satisfied for k = 3. Indeed, a direct calculation of the limit for k = 3 gives the value A = 1/2: Note that for k > 3 we get an infinite limit, and for , the limit will be equal to zero.

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Kolodzei Alexander Vladimirovich Asymptotic properties of goodness-of-fit criteria for testing hypotheses in a selection scheme without replacement, based on the filling of cells in a generalized allocation scheme: dissertation ... candidate of physical and mathematical sciences: 01.01.05.- Moscow, 2006.- 110 p.: ill. RSL OD, 61 07-1/496

Introduction

1 Entropy and Information Distance 36

1.1 Basic definitions and symbols 36

1.2 Entropy of Discrete Distributions with Bounded Expectation 39

1.3 Logarithmic generalized metric on a set of discrete distributions 43

1.4 Compactness of functions of a countable set of arguments. 46

1.5 Continuity of Kullback-Leibler-Sanov information distance 49

1.6 Conclusions 67

2 Large deviation probabilities 68

2.1 Probabilities of large deviations of functions from the number of cells with a given filling 68

2.1.1 Local limit theorem 68

2.1.2 Integral limit theorem 70

2.1.3 Information distance and large deviation probabilities of separable statistics 75

2.2 Large deviation probabilities of separable statistics that do not satisfy the Cramer condition 81

2.3 Conclusions 90

3 Asymptotic properties of goodness-of-fit tests 92

3.1 Acceptance criteria for the no-return selection scheme. 92

3.2 Asymptotic relative efficiency of goodness of fit tests 94

3.3 Criteria based on the number of cells in generalized layouts 95

3.4 Conclusions 98

Conclusion 99

Literature 103

Introduction to work

The object of research and the relevance of the topic. In the theory of statistical analysis of discrete sequences, a special place is occupied by goodness-of-fit tests for testing the possibly complex null hypothesis, which is that for a random sequence pQ)?=i such that

Хі Є Ім,і= 1,...,n, Ім = (o, i,..., M), for any і = 1,..., n, and for any k Є їм the probability of the event (Хі = k) does not depend on r. This means that the sequence (Xi)f = 1 is in some sense stationary.

In a number applied tasks as a sequence (Х() =1 we consider the sequence of colors of balls when choosing without returning to exhaustion from the urn containing rik - 1 > 0 balls of color k, k .,pd/ - 1) Let the urn contain n - 1 balls, m n-l= (n fc -l).

Denote by r(k) _ r(fc) r(fc) the sequence of numbers of balls of color k in the sample. Consider the sequence h" = (^,...,)). M fc) =ri fc) , ^ = ^-^ = 2,...,^-1, _ (fc)

The sequence h^ is defined by means of the distances between the places of adjacent balls of color k in such a way that *Ф = n.

The set of sequences h(fc) for all k Є їm uniquely determines the sequence the sequence of colors of balls is uniquely determined by the sequence h() of the distances between the places of neighboring balls of the same fixed color.Let an urn containing n - 1 balls of two different colors contain N - 1 balls of color 0. One can establish a one-to-one correspondence between the set M(N-l,n - N) and a set of 9 \ Nі m vectors h(n, N) = (hi,..., /i#) with positive integer components such that

The set 9\n,m corresponds to the set of all distinct partitions of a positive integer n into N ordered summands.

Having given some probability distribution on the set of vectors 9H n g, we obtain the corresponding probability distribution on the set Wl(N - l,n - N). The set Y\n,s is a subset of the set 2J n ,iv of vectors with non-negative integer components satisfying (0.1). As probability distributions on the set of vectors JZ p d in the dissertation work, distributions of the form

P(x, N) = (r t..., r N)) = P(& = rn, u = 1,..., N\ & = n), (0.2) where 6 > , n - independent non-negative integer random variables.

Distributions of the form (0.2) in /24/ were called generalized schemes for placing n particles in N cells. In particular, if the random variables h..., n in (0.2) are distributed according to the Poisson laws with the parameters Ai,...,Alg, respectively, then the vector h(n,N) has a polynomial distribution with the probabilities of outcomes

Pu \u003d m--~t~\u003e ^ \u003d 1,---, ^-

Li + ... + l^

If the random variables i> >&v in (0.2) are equally distributed according to the geometric law V(Zi = k)= P k - 1 (l-p),k=l,2,..., where p is any in the interval 0

As noted in /14/,/38/, a special place in testing hypotheses about the distribution of frequency vectors h(n, N) = (hi,..., h^) in generalized schemes for placing n particles in N cells is occupied by criteria constructed on the basis of statistics of the form

Фк "%,%..;$, (0.4) where /j/, v = 1,2,... and φ are some real-valued functions,

Mg \u003d E 1 (K \u003d g), g \u003d 0.1, .... 1 / \u003d 1

The quantities //r in /27/ were called the number of cells containing exactly r particles.

Statistics of the form (0.3) in /30/ are called separable (additively separable) statistics. If the functions /n in (0.3) do not depend on u, then such statistics were called in /31/ symmetric separable statistics.

For any r, the statistic fx r is a symmetric separable statistic. From equality

DM = DFg (0.5) it follows that the class of symmetric separable statistics in h u coincides with the class of linear functions in fi r . Moreover, the class of functions of the form (0.4) is wider than the class of symmetric separable statistics.

H 0 = (R0(n, N0) is a sequence of simple null hypotheses that the distribution of the vector h(n, N) is (0.2), where the random variables i,..., n, and (0.2) are identically distributed and P(ti = k)=p k ,k = 0,l,2,..., parameters n, N change in the central region.

Consider some РЄ (0,1) and a sequence of, generally speaking, complex alternatives n = (H(n,N)) such that there exists a n

P(Fm > OpAR)) >: 0-We will reject the hypothesis Hq(ti,N) if fm > a w m((3). If there exists a limit jim ~1nP(0lg > a n, N (P)) = Sh ), where the probability for each N is calculated under the hypothesis #o(n,iV), then the value j (fi,lcl) is called in /38/ the criterion index φ at the point (/?,Н). The last limit may, generally speaking, not exist. Therefore, in the dissertation work, in addition to the criterion index, the value lim (_IlnP(tor > a N (J3))) =іф(Р,П) is considered, which, by analogy, was called by the author of the dissertation work the lower index of the criterion f at the point (/3,Н) . Here and below, lim adg, lim a# jV-yuo LG-yuo mean, respectively, the lower and upper limits of the sequence (odr) as N -> syu,

If a criterion index exists, then the subscript of the criterion matches it. The subscript of the criterion always exists. How more value criterion index (lower criterion index), the better the statistical criterion in the considered sense. In /38/ the problem of constructing goodness-of-fit criteria for generalized layouts with highest value criterion index in the class of criteria that reject the hypothesis Ho(n, N) for where m > 0 is some fixed number, the sequence of constants eg is selected based on the given value of the power of the criterion with a sequence of alternatives, ft is a real function of m + 1 arguments.

The criteria indices are determined by the probabilities of large deviations. As shown in /38/, the rough (up to logarithmic equivalence) asymptotics of the probabilities of large deviations of separable statistics when the Cramer condition for the random variable /() is satisfied is determined by the corresponding Kullback-Leibler-Sanov information distance (the random variable μ satisfies the Cramer condition , if for some # > 0 the moment generating function Me f7? is finite in the interval \t\

The question of the probabilities of large deviations of statistics from an unbounded number fi r , as well as arbitrary separable statistics that do not satisfy the Cramer condition, remained open. This did not make it possible to finally solve the problem of constructing criteria for testing hypotheses in generalized allocation schemes with the highest rate of convergence to zero for the probability of an error of the first kind in the case of converging alternatives in the class of criteria based on statistics of the form (0.4). The relevance of the dissertation research is determined by the need to complete the solution of this problem.

The purpose of the dissertation work is to construct goodness-of-fit criteria with the highest value of the criterion index (lower index of the criterion) for testing hypotheses in the selection scheme without returning in the class of criteria that reject the hypothesis W(n, N) at 0(iv"iv"-""" o """)>CiV "(0" 7) where φ is a function of a countable number of arguments, and the parameters n, N change in the central region.

In accordance with the purpose of the study, the following tasks were set: to investigate the properties of entropy and informational distance of Kullback - Leibler - Sanov for discrete distributions with a countable number of outcomes; study the probabilities of large deviations of statistics of the form (0.4); study the probabilities of large deviations of symmetric separable statistics (0.3) that do not satisfy the Cramer condition; - find such a statistic that the agreement criterion built on its basis for testing hypotheses in generalized allocation schemes has the largest index value in the class of criteria of the form (0.7).

Scientific novelty: the concept of a generalized metric is given - a function that admits infinite values ​​and satisfies the axioms of identity, symmetry and triangle inequality. A generalized metric is found and sets are indicated on which the functions of entropy and information distance, given on a family of discrete distributions with a countable number of outcomes, are continuous in this metric; in the generalized allocation scheme, a rough (up to logarithmic equivalence) asymptotics is found for the probabilities of large deviations of statistics of the form (0.4) satisfying the corresponding form of Cramer's condition; in the generalized allocation scheme, a rough (up to logarithmic equivalence) asymptotics is found for the probabilities of large deviations of symmetric separable statistics that do not satisfy the Cramer condition; in the class of criteria of the form (0.7), a criterion with the largest value of the criterion index is constructed.

Scientific and practical value. In the paper, a number of questions about the behavior of large deviation probabilities in generalized allocation schemes are solved. The results obtained can be used in educational process in the specialties of mathematical statistics and information theory, in the study of statistical procedures for the analysis of discrete sequences and were used in /3/, /21/ when justifying the security of one class of information systems. Propositions to be defended: reduction of the problem of checking, using a single sequence of colors of balls, the hypothesis that this sequence was obtained as a result of a choice without replacement until the exhaustion of balls from the urn containing balls of two colors, and each such choice has the same probability, to the construction of goodness-of-fit criteria to test hypotheses in the corresponding generalized layout; continuity of the Kullback-Leibler-Sanov entropy and information distance functions on an infinite-dimensional simplex with the introduced logarithmic generalized metric; a theorem on the rough (up to logarithmic equivalence) asymptotics of the probabilities of large deviations of symmetric separable statistics that do not satisfy the Cramer condition in the generalized allocation scheme in the seven exionential case; a theorem on the rough (up to logarithmic equivalence) asymptotics of the probabilities of large deviations for statistics of the form (0.4); - construction of an agreement criterion for testing hypotheses in generalized allocation schemes with the largest index value in the class of criteria of the form (0.7).

Approbation of work. The results were reported at the seminars of the Department of Discrete Mathematics of the Mathematical Institute. V. A. Steklov RAS, Department of Information Security ITMiVT them. S. A. Lebedev RAS and at: the fifth All-Russian Symposium on Applied and Industrial Mathematics. Spring session, Kislovodsk, May 2 - 8, 2004; the sixth International Petrozavodsk conference "Probabilistic Methods in Discrete Mathematics" June 10 - 16, 2004; second International Conference"Information Systems and Technologies (IST" 2004)", Minsk, November 8 - 10, 2004;

International conference "Modern Problems and new Trends in Probability Theory", Chernivtsi, Ukraine, June 19 - 26, 2005.

The main results of the work were used in the research work "Apologia", carried out by ITMiVT RAS. S. A. Lebedev in the interests of the Federal Service for Technical and Export Control of the Russian Federation, and were included in the report on the implementation of the research stage /21/. Separate results of the dissertation were included in the research report "Development of mathematical problems of cryptography" of the Academy of Cryptography of the Russian Federation for 2004 /22/.

The author expresses his deep gratitude to the scientific adviser, Doctor of Physical and Mathematical Sciences Ronzhin A.F. and scientific consultant, Doctor of Physical and Mathematical Sciences, Senior Researcher Knyazev A.V. Mathematical Sciences I. A. Kruglov for the attention shown to the work and a number of valuable remarks.

Structure and content of the work.

The first chapter investigates the properties of entropy and information distance for distributions on the set of non-negative integers.

In the first paragraph of the first chapter, the notation is introduced and the necessary definitions are given. In particular, they are used the following notation: x = (:ro,i, ---) - an infinite-dimensional vector with a countable number of components;

H(x) - -Ex^oXvlnx,; trunc m (x) = (x 0 ,x 1 ,...,x t,0,0,...); SI* = (x, x u > 0, u = 0,1,..., E~ o xn 0,v = 0,l,...,E? =Q x v = 1); fi 7 \u003d (x Є O, L 0 vx v \u003d 7); %] = (хЄП, Ео»х and

16 mі = e o ** v \ &c = Ue>1 | 5 є Q 7) o

It is clear that the set Vt corresponds to the family of probability distributions on the set of non-negative integers, P 7 - to the family of probability distributions on the set of non-negative integers with mathematical expectation

Оє(у) - (х eO,x v

In the second paragraph of the first chapter, we prove a theorem on the boundedness of the entropy of discrete distributions with bounded mathematical expectation.

Theorem 1. On the boundedness of the entropy of discrete distributions with bounded mathematical expectation. For any wbp 7

If x Є fi 7 corresponds to a geometric distribution with mathematical prediction 7 ; that is

7xn = (1-p)p\ v = 0,1,..., where p = --,

1 + 7 then the equality H(x) = F(1) holds.

The assertion of the theorem can be viewed as the result of a formal application of the method of conditional Lagrange multipliers in the case of an infinite number of variables. The theorem that the only distribution on the set (k, k + 1, k + 2,...) with a given mathematical expectation and maximum entropy is a geometric distribution with a given mathematical expectation is given (without proof) in /47/. The author, however, gave a rigorous proof.

In the third paragraph of the first chapter, a definition of a generalized metric is given - a metric that admits infinite values.

For x, y Є Гі, the function p(x, y) is defined as the minimum є > 0 with the property y v e~ e

If there is no such є, then it is assumed that p(x, y) = oo.

It is proved that the function p(x, y) is a generalized metric on the family of distributions on the set of non-negative integers, as well as on the entire set Ci*. Instead of e in the definition of the metric p(x, y), you can use any other positive number other than 1. The resulting metrics will differ by a multiplicative constant. Denote by J(x, y) the information distance

Here and below it is assumed that 0 In 0 = 0.01n ^ = 0. The information distance is defined for such x, y, that x v - 0 for all and such that y v = 0. If this condition is not satisfied, then we will assume J (S,y) = co. Let A C $1. Then we will denote J(Ay)="mU(x,y).

Let J(Jb,y) = 00.

In the fourth paragraph of the first chapter, a definition is given for the compactness of functions defined on the set Π*. The compactness of a function with a countable number of arguments means that, with any degree of accuracy, the value of the function can be approximated by the values ​​of this function at points where only a finite number of arguments are nonzero. The compactness of the entropy and information distance functions is proved.

For any 0

If, for some 0 0, the function \(x) = J(x, p) is compact on the set ^ 7 ] P 0 r (p).

In the fifth paragraph of the first chapter, the properties of the information distance given on an infinite-dimensional space are considered. Compared to the finite-dimensional case, the situation with the continuity of the information distance function changes qualitatively. It is shown that the information distance function is not continuous on the set Г2 in any of the metrics pi(,y)= E|z„-i/„|, (

00 \ 2 p 2 (x, y) = sup (x^-ij^.

The validity of the following inequalities for the functions of entropy H(x) and information distance J(x,p) is proved:

1. For any x, x "Є fi \ H (x) - H (x") \

2. If for some x, p є P there exists є > 0 such that x є O є (p), then for any X i Є Q \J(x,p) - J(x",p)\

From these inequalities, taking into account Theorem 1, it follows that the entropy and information distance functions are uniformly continuous on the corresponding subsets fi in the p(x,y) metric, namely,

For any 7 such that 0

If for some 70, 0

20 then for any 0 0 the function \p(x) = J(x t p) is uniformly continuous on the set П 7 ] П О є (р) in the metric р(х,у).

The definition of non-extremality of a function is given. The non-extremality condition means that the function does not have local extrema, or the function takes the same values ​​in local minima (local maxima). The non-extremality condition weakens the requirement that there are no local extrema. For example, the function sin x on the set of real numbers has local extrema, but satisfies the condition of non-extremality.

Let for some 7 > 0, the area A is given by the condition

А = (хЄЇ1 1 ,Ф(х) >а), (0.9) where Ф(х) is a real-valued function, a is some real constant, inf Ф(х)

And 3y, the question arose, under what conditions „a f „ f with u_ „ parameters n, N in the central region, ^ -> 7, for all sufficiently large values ​​of them there are such non-negative integers ko, k\, ..., k n, which is ko + hi + ... + k n = N,

21 k\ + 2/... + nk n - N

Kq k \ k n . ^"iv"-"iv" 0 " 0 "-")>a -

It is proved that for this it suffices to require that the function φ be non-extremal, compact and continuous in the metric p(x, y), and also that for at least one point x satisfying (0.9), for some є > 0 there exists a finite moment of degree 1 + є Ml + = і 1+є x and 0 for any u = 0.1,....

In the second chapter, we study the rough (up to logarithmic equivalence) asymptotics of the probability of large deviations of functions from D = (fio,..., n, 0,...) - the number of cells with a given filling in the central region of the parameters N,n . The rough asymptotics of the probabilities of large deviations is sufficient to study the indices of the goodness of fit tests.

Let the random variables ^ in (0.2) be identically distributed and

Р(Сі = k)=р b k = 0.1,... > P(z) - generating function of random variable i - converges in a circle of radius 1

22 Denote p(.) = (p(ad = o), Pn) = i),...).

If there exists a solution z 1 of the equation

M(*) = 7, then it is unique /38/. Everywhere below we will assume that Pjfc>0,fc = 0,l,....

In the first paragraph of the first paragraph of the second chapter, there is an asymptotics of the logarithms of the probabilities of the form

The following theorem is proved.

Theorem 2. A rough local theorem on the probabilities of large deviations. Let n, N - * w so that - -> 7> 0

The statement of the theorem follows directly from the formula for the joint distribution /to, A*b / in /26/ and the following estimate: if non-negative integer values ​​fii,fi2,/ satisfy the condition /І1 + 2// 2 + ... + 71/ = 71, then the number of non-zero values ​​among them is 0(l/n). This is a rough estimate that does not claim to be new. The number of non-zero r in generalized layouts does not exceed the value of the maximum filling of cells, which in the central region with a probability tending to 1 does not exceed the value 0(\np) /25/,/27/. Nevertheless, the resulting estimate 0(y/n) is satisfied with probability 1, and it is sufficient to obtain a rough asymptotics.

In the second paragraph of the first paragraph of the second chapter, the value of the limit is found where adz is a sequence of real numbers converging to some a Є R, φ(x) is a real-valued function. The following theorem is proved.

Theorem 3. A rough integral theorem on the probabilities of large deviations. Let the conditions of Theorem 2 be satisfied, for some r > 0, (> 0 the real function φ(x) is compact, uniformly continuous in the metric p on the set

A = 0 rH (p(r 1))np n] and satisfies the condition of non-extremality on the set r2 7 . If for some constant a such that inf φ(x)

24 there is a vector p a fi 7 P 0 r (p(z 7)) ; such that

Ф(pа) > a J(( (x) >а,хЄ П 7 ),р(2; 7)) = J(p a ,p(^y)), mo for any sequence a^ converging to a, ^-^\nP(φ(φ, ^,...)>a m) = Pr a,p(r,)). (0.11)

Under additional restrictions on the function φ(x), the information distance J(pa, P(zy)) in (2.3) can be calculated more specifically. Namely, the following theorem is true. Theorem 4. Information distance. Let for some 0

Whether some r > 0, C > 0 the real function φ(x) and its first-order partial derivatives are compact and uniformly continuous in the generalized metric p(x, y) on the set

A = 0 r (p)PP bl] , there exist T > 0, R > 0 such that for all \t\ 0 p v v 1+ z u exp(i--φ(x))

0(p(gaL)) = a, / x X v \Z,t) T, u= oX LJ (Z,t)

Then p(z a, t a) Є ft, u J ((z Є L, 0 (x) = a), p) = J (p (z a , t a), p) d _ 9 = 7111 + t a "-^ OFaL)) - In 2Wexp( a --0(p(r a, i a))). j/=0 CnEi/ ^_o CX(/

If the function φ(x) is a linear function, and the function fix) is defined using equality (0.5), then condition (0.12) becomes the Cramer condition for the random variable f(,(z)). Condition (0.13) is a form of condition (0.10) and is used to prove the presence in domains of the form (x Є T2, φ(x) > a) of at least one point from 0(n, N) for all sufficiently large n, N.

Let v ()(n,iV) = (/i,...,/ijv) be the frequency vector in the generalized allocation scheme (0.2). As a consequence of Theorems 3 and 4, the following theorem is formulated.

Theorem 5. A rough integral theorem on the probabilities of large deviations of symmetric separable statistics in a generalized allocation scheme.

Let n, N -> w so that jfr - 7» 0 0, R > 0 such that for all \t\ Then for any sequence a# converging to a, 1 i iv =

This theorem was first proved by AF Ronzhin in /38/ using the saddle point method.

In the second section of the second chapter, we study the probabilities of large deviations of separable statistics in generalized cxj^iax arrangements in the case of non-fulfillment of the Cramer condition for the random variable /((z)). Cramer's condition for the random variable f(,(z)) is not satisfied, in particular, if (z) is a Poisson random variable, and /(x) = x 2 . Note that Cramer's condition for the separable statistics themselves in generalized allocation schemes is always satisfied, since for any fixed n, N the number possible outcomes in these charts, of course.

As noted in /2/, if the Cramer condition is not satisfied, then to find the asymptotics of the probabilities of large deviations of sums of identically distributed random variables fulfillment of additional f conditions for a correct change in the distribution of the term is required. In the paper (the case is considered that corresponds to the fulfillment of condition (3) in /2/, that is, the seven-exponential case. Let P(i = k) > 0 for all

28 k = 0.1,... and the function p(k) = -\nP(t = k), can be extended to a function of continuous argument - a regularly varying function of order p, 0 oo P(tx) , r v P(t )

Let the function f(x) for sufficiently large values ​​of the argument be a positive, strictly increasing, regularly varying function of order q > 1, on the rest of the real axis

Then s. V. /(i) has moments of any order and does not satisfy Cramer's condition, ip(x) = o(x) as x -> oo, and the following theorem holds. ^p does not increase monotonically, n, N --> oo so that jf - A, 0 b(z\), where b(z) = M/(1(2)), there exists a limit l(n,l)) > cN] = "(c ~ b(zx))l b""ї

It follows from Theorem 6 that, if the Cramer condition is not satisfied, the limit (^ lim ~\nP(L N (h(n,N)) > cN) = 0, "" Dv

L/-too iV and that proves the validity of the hypothesis stated in /39/. Thus, the value of the index of the goodness-of-fit criterion in generalized allocation schemes -^ when Cramer's condition is not satisfied, is always equal to zero. In this case, in the class of criteria, when the Cramer condition is satisfied, criteria with a non-zero index value are constructed. From this we can conclude that using criteria whose statistics do not satisfy the Cramer condition, for example, the chi-square test in a polynomial scheme, to construct goodness-of-fit tests for testing hypotheses with non-approaching alternatives is asymptotically inefficient in this sense. A similar conclusion was made in /54/ based on the results of comparing the chi-square statistics and the maximum likelihood ratio in a polynomial scheme.

In the third chapter, we solve the problem of constructing goodness-of-fit criteria with the highest value of the criterion index (the largest value of the lower index of the criterion) for testing hypotheses in generalized layouts. Based on the results of the first and second chapters on the properties of entropy functions, information distance, and probabilities of large deviations, in the third chapter, a function of the form (0.4) is found such that the goodness-of-fit criterion built on its basis has the largest value of the exact lower index in the class of criteria under consideration. The following theorem is proved. Theorem 7. On the existence of an index. Let the conditions of Theorem 3 be satisfied, 0 ,... is a sequence of alternative distributions, 0^(/3, iV) is the maximum number for which, under the hypothesis Н Р (lo, the inequality

P(φ(^^,...) > a φ (P, M)) > (3, there exists a limit there is an index of the criterion f

3ff,K) = 3((φ(x) >a,xe 3D.P^)).

At the same time, sph(0,th)N NP(e(2 7) = fc)"

The Conclusion outlines the results obtained in their relationship with the general goal and specific tasks set in the dissertation, formulates conclusions based on the results of the dissertation research, indicates the scientific novelty, theoretical and practical value of the work, as well as specific scientific problems that have been identified by the author and the solution of which seems relevant. .

A brief review of the literature on the research topic.

The dissertation work considers the problem of constructing goodness of fit criteria in generalized allocation schemes with the largest value of the criterion index in the class of functions of the form (0.4) with non-approaching alternatives.

Generalized allocation schemes were introduced by VF Kolchin in /24/. The values ​​fi r in the polynomial scheme were called the number of cells with r shots and were studied in detail in the monograph by V. F. Kolchin, B. A. Sevastyanov, V. P. Chistyakov /27/. The values ​​\і r in generalized layouts were studied by VF Kolchin in /25/,/26/. Statistics of the form (0.3) were first considered by Yu. I. Medvedev in /30/ and were called separable (additively separable) statistics. If the functions /„ in (0.3) do not depend on u, such statistics were called in /31/ symmetric separable statistics. The asymptotic behavior of the moments of separable statistics in generalized allocation schemes was obtained by GI Ivchenko in /9/. Limit theorems for a generalized allocation scheme were also considered in /23/. Reviews of the results of limit theorems and goodness of fit in discrete probabilistic schemes of type (0.2) were given by V. A. Ivanov, G. I. Ivchenko, Yu. I. Medvedev in /8/ and G. I. Ivchenko, Yu. I. Medvedev , A.F. Ronzhin in /14/. Goodness-of-fit criteria for generalized layouts were considered by A.F. Ronzhin in /38/.

Comparison of the properties of statistical tests in these works was carried out from the point of view of relative asymptotic efficiency. The case of approaching (contigual) hypotheses - efficiency in the sense of Pitman and non-converging hypotheses - efficiency in the sense of Bahadur, Hodges - Lehman and Chernov were considered. Connection between various types the relative effectiveness of statistical criteria is discussed, for example, in /49/. As follows from the results of Yu. I. Medvedev in /31/ on the distribution of decomposable statistics in a polynomial scheme, the test based on the chi-square statistic has the highest asymptotic power under converging hypotheses in the class of decomposable statistics on the frequencies of outcomes in a polynomial scheme. This result was generalized by A.F. Ronzhin for schemes of type (0.2) in /38/. II Viktorova and VP Chistyakov in /4/ constructed an optimal criterion for a polynomial scheme in the class of linear functions of fi r . A. F. Ronzhin in /38/ constructed a criterion that, in the case of a sequence of alternatives not approaching the null hypothesis, minimizes the logarithmic rate of the probability of an error of the first kind tending to zero in the class of statistics of the form (0.6). A comparison of the relative performance of the chi-square statistic and the maximum likelihood ratio for converging and non-converging hypotheses was made in /54/. In the dissertation work, the case of non-approaching hypotheses was considered. The study of the relative statistical efficiency of criteria under nonconverging hypotheses requires the study of the probabilities of superlarge deviations - of the order of 0(y/n). For the first time such a problem for a polynomial distribution with a fixed number of outcomes was solved by IN Sanov in /40/. The asymptotic optimality of goodness-of-fit criteria for testing simple and complex hypotheses for a polynomial distribution in the case of a finite number of outcomes with non-approaching alternatives was considered in /48/. Properties of the information distance were previously considered by Kullback, Leibler /29/,/53/ and I. II. Sanov /40/, as well as Heffding /48/. In these papers, the continuity of the information distance was considered on finite-dimensional spaces in the Euclidean metric. The author also considered a sequence of spaces with increasing dimension, for example, in the work of Yu. V. Prokhorov /37/ or in the work of V. I. Bogachev, A. V. Kolesnikov /1/. Rough (up to logarithmic equivalence) theorems on the probabilities of large deviations of separable statistics in generalized allocation schemes under Cramer's condition were obtained by AF Roizhin in /38/. A. N. Timashev in /42/,/43/ obtained exact (up to equivalence) multidimensional integral and local limit theorems on the probabilities of large deviations of the vector fir^n, N),..., fi rs (n,N) , where s, гі,..., r s - fixed integers,

Statistical problems of testing hypotheses and estimating parameters in a selection scheme without replacement in a slightly different formulation were considered by G. I. Ivchenko, V. V. Levin, E. E. Timonina /10/, /15/, where estimation problems were solved for a finite population, when the number of its elements is an unknown value, the asymptotic normality of multivariate S-statistics from s independent samples in a selection scheme without replacement was proved. The problem of studying random variables associated with repetitions in sequences of independent trials was studied by A. M. Zubkov, V. G. Mikhailov, A. M. Shoitov in /6/, /7/, /32/, /33/, / 34/. Analysis of the main statistical problems of estimation and testing of hypotheses in the framework general model Markov-Poya was carried out by G. I. Ivchenko, Yu. I. Medvedev in /13/, the probabilistic analysis of which was given in /11/. A method for specifying nonequiprobable measures on a set of combinatorial objects that is not reducible to a generalized allocation scheme (0.2) was described in GI Ivchenko, Yu. I. Medvedev /12/. A number of problems in probability theory, in which the answer can be obtained as a result of calculations using recursive formulas, is indicated by AM Zubkov in /5/.

The inequalities for the entropy of discrete distributions were obtained in /50/ (cited in the abstract of A. M. Zubkov in RZhMat). If (p n )Lo is a probability distribution,

Pn = E Pk, k=n A = supp^Pn+i

I + (In -f-) (X Rp - P p + 1)

Рп= (x f 1)n+v n>Q. (0.15)

Note that the extremal distribution (0.15) is a geometric distribution with the expectation A, and the function F(X) of the parameter (0.14) coincides with the function of the expectation in Theorem 1.

Entropy of Discrete Distributions with Bounded Expectation

If a criterion index exists, then the subscript of the criterion matches it. The subscript of the criterion always exists. The greater the value of the criterion index (lower index of the criterion), the better the statistical criterion in the considered sense. In /38/, the problem of constructing goodness-of-fit criteria for generalized layouts with the highest value of the criterion index in the class of criteria that reject the hypothesis Ho(n,N) was solved for where m 0 is some fixed number, the sequence of constants eg is selected based on the given value the power of the criterion for a sequence of alternatives, ft is a real function of m + 1 arguments.

The criteria indices are determined by the probabilities of large deviations. As shown in /38/, the rough (up to logarithmic equivalence) asymptotics of the probabilities of large deviations of separable statistics when the Cramer condition for the random variable /() is satisfied is determined by the corresponding Kullback-Leibler-Sanov information distance (the random variable μ satisfies the Cramer condition , if for some # 0 the moment generating function Mef7? is finite in the interval \t\ H /28/).

The question of the probabilities of large deviations of statistics from an unbounded number fir, as well as of arbitrary separable statistics that do not satisfy the Cramer condition, remained open. This did not make it possible to finally solve the problem of constructing criteria for testing hypotheses in generalized allocation schemes with the highest rate of convergence to zero for the probability of an error of the first kind in the case of converging alternatives in the class of criteria based on statistics of the form (0.4). The relevance of the dissertation research is determined by the need to complete the solution of this problem.

The aim of the dissertation work is to construct goodness-of-fit criteria with the highest value of the criterion index (lower index of the criterion) for testing hypotheses in the selection scheme without recurrence in the class of criteria that reject the hypothesis W(n, N) for where φ is a function of a countable number of arguments, and the parameters n, N change in the central region. In accordance with the purpose of the study, the following tasks were set: - to investigate the properties of entropy and the Kullback - Leibler - Sanov information distance for discrete distributions with a countable number of outcomes; - investigate the probabilities of large deviations of statistics of the form (0.4); - investigate the probabilities of large deviations of symmetric separable statistics (0.3) that do not satisfy the Cramer condition; - find such a statistic that the agreement criterion built on its basis for testing hypotheses in generalized allocation schemes has the largest index value in the class of criteria of the form (0.7). Scientific novelty: - the concept of a generalized metric is given - a function that admits infinite values ​​and satisfies the axioms of identity, symmetry and triangle inequality. A generalized metric is found and sets are indicated on which the functions of entropy and information distance, given on a family of discrete distributions with a countable number of outcomes, are continuous in this metric; - in the generalized allocation scheme, a rough (up to logarithmic equivalence) asymptotics is found for the probabilities of large deviations of statistics of the form (0.4) satisfying the corresponding form of the Cramer condition; - in the generalized allocation scheme, a rough (up to logarithmic equivalence) asymptotics is found for the probabilities of large deviations of symmetric separable statistics that do not satisfy the Cramer condition; - in the class of criteria of the form (0.7), a criterion with the largest value of the criterion index is built. Scientific and practical value. In the paper, a number of questions about the behavior of large deviation probabilities in generalized allocation schemes are solved. The results obtained can be used in the educational process in the specialties of mathematical statistics and information theory, in the study of statistical procedures for the analysis of discrete sequences and were used in /3/, /21/ when justifying the security of one class of information systems. Provisions put forward for defense: - reduction of the problem of checking, using a single sequence of colors of balls, the hypothesis that this sequence was obtained as a result of a choice without replacement until the exhaustion of balls from an urn containing balls of two colors, and each such choice has the same probability, to the construction of criteria agreement to test hypotheses in the corresponding generalized layout; - continuity of the functions of entropy and Kullback - Leibler - Sanov information distance on an infinite-dimensional simplex with the introduced logarithmic generalized metric; - a theorem on the rough (up to logarithmic equivalence) asymptotics of the probabilities of large deviations of symmetric separable statistics that do not satisfy the Cramer condition in the generalized allocation scheme in the seven exionential case;

Continuity of Kullback-Leibler-Sanov Information Distance

Generalized allocation schemes were introduced by VF Kolchin in /24/. The values ​​fir in the polynomial scheme were called the number of cells with r shots and were studied in detail in the monograph by V. F. Kolchin, B. A. Sevastyanov, V. P. Chistyakov /27/. The values ​​\іr in generalized layouts were studied by VF Kolchin in /25/,/26/. Statistics of the form (0.3) were first considered by Yu. I. Medvedev in /30/ and were called separable (additively separable) statistics. If the functions /„ in (0.3) do not depend on u, such statistics were called in /31/ symmetric separable statistics. The asymptotic behavior of the moments of separable statistics in generalized allocation schemes was obtained by GI Ivchenko in /9/. Limit theorems for a generalized allocation scheme were also considered in /23/. Reviews of the results of limit theorems and goodness of fit in discrete probabilistic schemes of type (0.2) were given by V. A. Ivanov, G. I. Ivchenko, Yu. I. Medvedev in /8/ and G. I. Ivchenko, Yu. I. Medvedev , A.F. Ronzhin in /14/. Goodness-of-fit criteria for generalized layouts were considered by A.F. Ronzhin in /38/.

Comparison of the properties of statistical tests in these works was carried out from the point of view of relative asymptotic efficiency. The case of approaching (contigual) hypotheses - efficiency in the sense of Pitman and non-converging hypotheses - efficiency in the sense of Bahadur, Hodges - Lehman and Chernov were considered. The relationship between different types of relative performance of statistical tests is discussed, for example, in /49/. As follows from the results of Yu. I. Medvedev in /31/ on the distribution of separable statistics in a polynomial scheme, the test based on the chi-square statistic has the highest asymptotic power under converging hypotheses in the class of separable statistics on the frequencies of outcomes in a polynomial scheme. This result was generalized by A.F. Ronzhin for schemes of type (0.2) in /38/. II Viktorova and VP Chistyakov in /4/ constructed an optimal criterion for a polynomial scheme in the class of linear functions of fir. A. F. Ronzhin in /38/ constructed a criterion that, in the case of a sequence of alternatives not approaching the null hypothesis, minimizes the logarithmic rate of the probability of an error of the first kind tending to zero in the class of statistics of the form (0.6). A comparison of the relative performance of the chi-square statistic and the maximum likelihood ratio for converging and non-converging hypotheses was made in /54/. In the dissertation work, the case of non-approaching hypotheses was considered. The study of the relative statistical efficiency of criteria under nonconverging hypotheses requires the study of the probabilities of superlarge deviations - of the order of 0(y/n). For the first time such a problem for a polynomial distribution with a fixed number of outcomes was solved by IN Sanov in /40/. The asymptotic optimality of goodness-of-fit criteria for testing simple and complex hypotheses for a polynomial distribution in the case of a finite number of outcomes with non-approaching alternatives was considered in /48/. Properties of the information distance were previously considered by Kullback, Leibler /29/,/53/ and I. II. Sanov /40/, as well as Heffding /48/. In these papers, the continuity of the information distance was considered on finite-dimensional spaces in the Euclidean metric. The author also considered a sequence of spaces with increasing dimension, for example, in the work of Yu. V. Prokhorov /37/ or in the work of V. I. Bogachev, A. V. Kolesnikov /1/. Rough (up to logarithmic equivalence) theorems on the probabilities of large deviations of separable statistics in generalized allocation schemes under the Cramer condition were obtained by A. F. Roizhin in /38/. A. N. Timashev in /42/,/43/ obtained exact (up to equivalence) multidimensional integral and local limit theorems on the probabilities of large deviations of a vector

The study of the probabilities of large deviations when Cramer's condition is not met for the case of independent random variables was carried out in the works of A. V. Nagaev /35/. The method of conjugate distributions is described by Feller /45/.

Statistical problems of testing hypotheses and estimating parameters in a selection scheme without replacement in a slightly different formulation were considered by G. I. Ivchenko, V. V. Levin, E. E. Timonina /10/, /15/, where estimation problems were solved for a finite population, when the number of its elements is an unknown value, the asymptotic normality of multivariate S-statistics from s independent samples in a selection scheme without replacement was proved. The problem of studying random variables associated with repetitions in sequences of independent trials was studied by A. M. Zubkov, V. G. Mikhailov, A. M. Shoitov in /6/, /7/, /32/, /33/, /34/ . The analysis of the main statistical problems of estimation and testing of hypotheses in the framework of the general Markov-Poya model was carried out by G. I. Ivchenko, Yu. I. Medvedev in /13/, the probabilistic analysis of which was given in /11/. A method for specifying nonequiprobable measures on a set of combinatorial objects that is not reducible to a generalized allocation scheme (0.2) was described in GI Ivchenko, Yu. I. Medvedev /12/. A number of problems in probability theory, in which the answer can be obtained as a result of calculations using recursive formulas, is indicated by AM Zubkov in /5/.

Information distance and probabilities of large deviations of separable statistics

When Cramer's condition is not satisfied, large deviations of separable statistics in the generalized allocation scheme in the considered seven-exponential case are determined by the probability of deviation of one independent term. When Cramer's condition is satisfied, this, as emphasized in /39/, is not the case. Remark 10. The function φ(x) is such that the mathematical expectation Ee (A) is finite at 0 t 1 and infinite at t 1. Remark 11. For separable statistics that do not satisfy the Cramer condition, the limit (2.14) is equal to 0, which proves the validity of the conjecture expressed in /39/. Remark 12. For the chi-square statistic in the polynomial scheme for n, ./V - co such that - A, it follows directly from the theorem that This result was obtained directly in /54/. In this chapter, in the central range of the parameters of generalized schemes for distributing particles over cells, rough (up to logarithmic equivalence) asymptotics of the probabilities of large deviations of additively separable statistics from cell filling and functions of the number of cells with a given filling were found.

If Cramer's condition is satisfied, then the rough asymptotics of the probabilities of large deviations is determined by the rough asymptotics of the probabilities of falling into a sequence of points with rational coordinates converging in the above sense to the point at which the extremum of the corresponding information distance is reached.

The seven-exponential case of non-fulfillment of the Cramer condition for the random variables f(i),..., f(x) was considered, where b, x are independent random variables generating the generalized partitioning scheme (0.2), f(k) is a function in definition of a symmetric additively separable statistic in (0.3). That is, it was assumed that the functions p(k) = - lnP(i = k) and f(k) can be extended to regularly varying functions of a continuous argument of the order p 0 and q 0, respectively, and p q . It turned out that the main contribution to the rough asymptotics of the probabilities of large deviations of separable statistics in generalized allocation schemes is similarly made by the rough asymptotics of the allocation probability to the corresponding sequence of points. It is interesting to note that earlier the theorem on the probabilities of large deviations for separable statistics was proved using the saddle point method, with the main contribution to the asymptotics being made by a single saddle point. The case remained unexplored when, if the Cramer condition is not satisfied, the 2-kN condition is not satisfied.

If Cramer's condition is not satisfied, then the indicated condition may not be satisfied only in the case of p 1. As follows directly from the logarithm of the corresponding probability distribution, for the Poisson distribution and the geometric distribution p=1. From the result on the asymptotics of the probabilities of large deviations when the Cramer condition is not satisfied, we can conclude that criteria whose statistics do not satisfy the Cramer condition have a significantly lower rate of convergence to zero of the probabilities of errors of the second kind for a fixed probability of error of the first kind and non-approaching alternatives compared to the criteria whose statistics satisfy the Cramer condition. Let an urn containing N - 1 1 white un-JV 1 black balls be chosen without replacement until it is exhausted. Let us relate the positions of the white balls in the choice 1 i\ ... r -i n - 1 with the sequence of distances hi,..., h between adjacent white balls as follows: Then hv l,v =1,... ,N,M EjLi i/ - n- Let us define a probability distribution on the set of vectors h = (hi,..., λg) by setting V(hv = rv,v = l,...,N) where i,... ,lg - independent non-negative integer random variables (r.v.), that is, consider the generalized allocation scheme (0.2). The distribution of the vector h depends on n,N, but the corresponding indices, where possible, will be omitted for ease of notation. Remark 14. If each of the (]) ways of choosing balls from an urn is assigned the same probability (\) mn for any r i,..., rg such that rn 1,u = l,...,N ,T,v=\ru = n, the probability that the distances between adjacent white balls in the choice take these values

Criteria based on the number of cells in generalized layouts

The purpose of the dissertation work was to construct goodness-of-fit criteria for testing hypotheses in a selection scheme without returning from an urn containing balls of 2 colors. The author decided to study statistics based on the frequency of distances between balls of the same color. In this formulation, the problem was reduced to the problem of testing hypotheses in a suitable generalized layout.

In the dissertation work were - investigated the properties of entropy and information distance of discrete distributions with an unlimited number of outcomes with a limited mathematical expectation; - a rough (up to logarithmic equivalence) asymptotics of the probabilities of large deviations of a wide class of statistics in a generalized allocation scheme has been obtained; - on the basis of the obtained results, a criterion function with the highest logarithmic rate of convergence to zero of the probability of an error of the first kind is constructed for a fixed probability of an error of the second kind and non-approaching alternatives; - It has been proved that statistics that do not satisfy the Cramer condition have a lower rate of tending to zero of the probabilities of large deviations compared to statistics that satisfy such a condition. The scientific novelty of the work is as follows. - the concept of a generalized metric is given - a function that admits infinite values ​​and satisfies the axioms of identity, symmetry and triangle inequality. A generalized metric is found and sets are indicated on which the functions of entropy and information distance, given on a family of discrete distributions with a countable number of outcomes, are continuous in this metric; - in the generalized allocation scheme, a rough (up to logarithmic equivalence) asymptotics is found for the probabilities of large deviations of statistics of the form (0.4) satisfying the corresponding form of the Cramer condition; - in the generalized allocation scheme, a rough (up to logarithmic equivalence) asymptotics is found for the probabilities of large deviations of symmetric separable statistics that do not satisfy the Cramer condition; - in the class of criteria of the form (0.7), a criterion with the largest value of the criterion index is built. In the paper, a number of questions about the behavior of large deviation probabilities in generalized allocation schemes are solved. The results obtained can be used in the educational process in the specialties of mathematical statistics and information theory, in the study of statistical procedures for the analysis of discrete sequences and were used in /3/, /21/ when justifying the security of one class of information systems. However, a number of questions remain open. The author limited himself to considering the central zone of change parameters n,N generalized schemes for arranging n particles in /V cells. If the carrier of the distribution of random variables generating the generalized allocation scheme (0.2) is not a set of the form r, r 4-1, r + 2,..., then when proving the continuity of the information distance function and studying the probabilities of large deviations, it is required to take into account the arithmetic structure of such carrier, which was not considered in the author's work. For the practical application of criteria built on the basis of the proposed function with the maximum value of the index, it is required to study its distribution both under the null hypothesis and under alternatives, including converging ones. It is also of interest to transfer the developed methods and to generalize the obtained results to other probabilistic schemes other than generalized allocation schemes. If //1,/ 2,-.. are the frequencies of the distances between the numbers of outcome 0 in the binomial scheme with the probabilities of outcomes рї 1 -POj, then it can be shown that in this case proved in /26/, it follows that the distribution (3.3), generally speaking, cannot be represented in the general case as a joint distribution of the values ​​of z in any generalized scheme for placing particles in cells. This distribution is a special case of distributions on the set of combinatorial objects introduced in /12/. It seems to be an urgent task to transfer the results of the dissertation work for generalized layouts to this case, which was discussed in /52/.